Question

If \(f(x)=x^{2}-2 x+3,\) find the value(s) of \(x\) so that \(f(x)=11\)

Short answer

The values of \( x \) are 4 and -2.

Step-by-step solution

- Set up the equation

Set the expression for the function equal to 11: \[ f(x) = x^2 - 2x + 3 = 11 \]

- Subtract 11 from both sides

Rewrite the equation by moving 11 to the left side: \[ x^2 - 2x + 3 - 11 = 0 \] Simplify the equation: \[ x^2 - 2x - 8 = 0 \]

- Factor the quadratic equation

Factor the quadratic equation \( x^2 - 2x - 8 \): \[ (x - 4)(x + 2) = 0 \]

- Solve for x

Set each factor equal to zero and solve for \( x \): \[ x - 4 = 0 \] \[ x = 4 \] and \[ x + 2 = 0 \] \[ x = -2 \]

Key concepts

factoring quadratics

Factoring quadratics is a method used to solve equations of the form \[ ax^2 + bx + c = 0 \]. This involves breaking down the quadratic equation into simpler binomials. In the given exercise, we have the quadratic equation \[ x^2 - 2x - 8 = 0 \]. Our goal is to express this equation in a factorized form.

For successful factoring, we need to find two numbers that multiply to \[ -8 \] (the constant term) and add up to \[ -2 \] (the linear coefficient). After analyzing, these two numbers are \[ -4 \] and \[ 2 \].

Therefore, the quadratic equation factors into:

\[ (x - 4)(x + 2) = 0 \]

This process might require some practice, but it's an effective way to break down complex quadratic equations.

solving equations

Once we have factored the quadratic equation, the next step is solving for the variable. Given the factors \[ (x - 4)(x + 2) = 0 \], we apply the zero-product property. This property states that if the product of two numbers is zero, at least one of the numbers must be zero.

So, we set each factor equal to zero and solve for \[ x \]:

\[ x - 4 = 0 \]
\[ x = 4 \]

and

\[ x + 2 = 0 \]
\[ x = -2 \]

The solutions to the quadratic equation are \[ x = 4 \] and \[ x = -2 \]. These values of \[ x \] satisfy the original equation, meaning they are the points where \[ f(x) = 11 \]. These steps illustrate how factoring and solving linear equations can be used together to find solutions.

function evaluation

To understand and solve function-related problems, one key step is function evaluation. In the given exercise, we have a function \[ f(x) = x^2 - 2x + 3 \]. We are asked to find the value(s) of \[ x \] such that \[ f(x) = 11 \].

The first step is to set the function equal to 11:
\[ x^2 - 2x + 3 = 11 \]. Next, by simplifying, we transform the equation:
\[ x^2 - 2x + 3 - 11 = 0 \]
\[ x^2 - 2x - 8 = 0 \].

This equation is then factored and solved to find the values of \[ x \].

Function evaluation plays an important role in linking algebraic manipulations to real-world scenarios. Here, by evaluating the function, we determine the specific points where the function achieves a given output. It helps understand the behavior of functions and their corresponding graphs.